Consider a polynomial function 
𝑓
(
𝑥
)
f(x) of degree 4 which intersects the X-axis at 
𝑥
=
2
,
𝑥
=
−
3
x=2,x=−3 and 
𝑥
=
−
4
x=−4. Moreover, 
𝑓
(
𝑥
)
<
0
f(x)<0 when 
𝑥
∈
(
1
,
2
)
x∈(1,2), and 
𝑓
(
𝑥
)
>
0
f(x)>0 when 
𝑥
∈
(
−
1
,
1
)
x∈(−1,1). Find out the equation of the polynomial 

𝑎
(
𝑥
−
2
)
2
(
𝑥
2
+
7
𝑥
+
12
)
,
𝑎
>
0
a(x−2) 
2
 (x 
2
 +7x+12),a>0

𝑎
(
𝑥
4
+
4
𝑥
3
−
7
𝑥
2
−
22
𝑥
+
24
)
,
𝑎
>
0
a(x 
4
 +4x 
3
 −7x 
2
 −22x+24),a>0

𝑎
(
𝑥
−
2
)
2
(
𝑥
2
+
2
𝑥
−
8
)
,
𝑎
>
0
a(x−2) 
2
 (x 
2
 +2x−8),a>0

𝑎
(
𝑥
4
−
5
𝑥
3
−
7
𝑥
2
−
50
𝑥
−
24
)
,
𝑎
>
0
a(x 
4
 −5x 
3
 −7x 
2
 −50x−24),a>0

To determine the correct polynomial \( f(x) \) that satisfies the given conditions, we need to consider the roots and the behavior of the polynomial.

The polynomial \( f(x) \) intersects the X-axis at \( x = 2 \), \( x = -3 \), and \( x = -4 \). Therefore, these values are roots of the polynomial. Given that \( f(x) \) is a polynomial of degree 4, one of these roots must be a repeated root to satisfy the degree condition.

Next, we examine the intervals where \( f(x) \) is positive or negative:
- \( f(x) < 0 \) for \( x \in (1, 2) \)
- \( f(x) > 0 \) for \( x \in (-1, 1) \)

This behavior suggests that \( x = 2 \) is a double root because the polynomial changes sign around \( x = 2 \). The other roots \( x = -3 \) and \( x = -4 \) are single roots.

Thus, we can express the polynomial \( f(x) \) in the form:
\[ f(x) = a(x - 2)^2(x + 3)(x + 4) \]
where \( a > 0 \) because the polynomial has to be positive in certain intervals.

Let's check which given options match this structure:

1. \( a(x-2)^2(x^2 + 7x + 12), a > 0 \)
2. \( a(x^4 + 4x^3 - 7x^2 - 22x + 24), a > 0 \)
3. \( a(x-2)^2(x^2 + 2x - 8), a > 0 \)
4. \( a(x^4 - 5x^3 - 7x^2 - 50x - 24), a > 0 \)

First, let's expand the factored form to see if it matches any of the given polynomials:
\[ f(x) = a(x - 2)^2(x + 3)(x + 4) \]

Let's expand \( (x + 3)(x + 4) \):
\[ (x + 3)(x + 4) = x^2 + 7x + 12 \]

So, the polynomial becomes:
\[ f(x) = a(x - 2)^2(x^2 + 7x + 12) \]

This matches the first option:
\[ a(x - 2)^2(x^2 + 7x + 12), \quad a > 0 \]

Thus, the equation of the polynomial is:
\[ \boxed{a(x - 2)^2(x^2 + 7x + 12), \, a > 0} \]

Would you like more details or have any questions about this solution?

Here are 5 numbered questions you can ask:
1. How do you determine the degree of a polynomial given its roots?
2. Can you explain how to expand polynomial expressions?
3. What is the significance of repeated roots in polynomial equations?
4. How do you analyze the sign changes of a polynomial function?
5. Can you provide more examples of finding polynomial equations given specific conditions?

**Tip:** Always check the degree and roots of a polynomial carefully to match it to a given form.

Discover how to determine a polynomial equation of degree 4 with specified roots and sign conditions. This problem explores polynomial functions, root analysis, and sign changes in polynomials. Suitable for advanced high school students.

Math Problem Statement

Consider a polynomial function 𝑓 ( 𝑥 ) f(x) of degree 4 which intersects the X-axis at 𝑥

2 , 𝑥

− 3 x=2,x=−3 and 𝑥

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